 
Summary: 44 n m TRANSACTIONS ON AUTOMATIC C O ~ O L ,VOL. AC25,NO. 1, FEBRUARY 1980
Maximal Order Reduction and Supremal
( A ,B)Invariant and Controllability Subspaces
P.J. ANTsAKLIs,MEMBER, IEEE
AbstractGiven the system {A,B,C,E) the supremal (A,B)invariant
and cwlmlhMily sobspaces are studied and thew dimensions are ex
pUcitly determined as functionsof the number of zeros and the degree of
the determinantof the interactor. This is done by solving the problem of
the maximalorder reduction via linear state feedback.
I. INTRODUC~~ON
The geometricapproach [9]has been used successfully in recent years
in theanalysis and synthesis of linear, multivariable,timeinvariant
systems. Among the key concepts of this approach are the concepts of
the supremal(A,B)iivariant and controllability subspacescontained in
a given subspace(often,thekernel of C), denoted by Y. and R*,
respectively. R* is a subspaceof vf and it has been shown by a number
of authors that dimvf =q +dimR* where q is the number of zeros of
the system considered. Note, however, that the actual dimension of R*
and consequentlyof vf are unknown. These dimensions are important
sincetheyhaveacriticaleffect on the dynamic controller structure in
